Lemma 5.1.2
The following conditions are equivalent for any unital -algebra
Proof:
()
If is an isometry, then by direct calculation we know that is a projection. Hence and is a unitary.
()
If for some projection . Then by the definition of murray von neumann equivalence, there exists an isometry which connects and . Thus and .But in this algebra, every isometry is a unitary, so , this shows that is equivalent to a proper subprojection of itself.
()
Suppose that and are two projections in such that and . Let be a partial isometry in which implements the equivalence between and . and . Put and from our assumption, so we get that . We conclude that and , but since all isometries are unitary, necessarily, proving that every projection is equivalent to some proper sub projection of itself.
() If holds, then the unit is a finite projection.
() Parse through what it means.
() Trivial.
()
Suppose that holds, and let be a left-invertible element in . find such that . Recall that if are self-adjoint elements in a unital -algebra , then and if then for each . Use these facts to obtain the inequality This shows that and the spectrum of is therefore contained in the interval , which in particular implies tat is invertible. Put , and observe that therefore is invertible by (2), and hence is invertible.
Warnings:
- A finite -algebra, unital or not, will always satisfy (3), that all projections are finite.
The converse is NOT TRUE
- A non-unital -algebra can be infinite and satisfy (3), have only finite projections.
- Lastly, a finite C-star algebra does not have to be stably finite.
NOTE:
Let be a properly infinite, unital -algebra, with nontrivial -group; the cuntz algebras for are examples of such algebras. Then by exercise 4.6(4), and hence therefore, are not ordered abelian groups.
Definition 5.16
An element in an ordered abelian group is called an order unit if for ever , there exists such that . In other words, you can wrangle any element with this order unit in the same way you can wrangle any number in the integers with 1. It makes sense I promise.
Proposition 5.1.7
If is a unital -algebra, then is an order unit for .
Proof:
Let be given. By the standard picture of of a unital -algebra we can find projections for some such that . Let denote the unit of , and write instead of . Then and belong to . Hence